International J.Math. Combin. Vol.4(2017), 84-90
Some Lower and Upper Bounds on the Third ABC Co-index Deepak S. Revankar1 , Priyanka S. Hande2 , Satish P. Hande3 and Vijay Teli3 1. Department of Mathematics, KLE, Dr. M. S. S. C. E. T., Belagavi - 590008, India 2. Department of Mathematics, KLS, Gogte Institute of Technology, Belagavi - 590008, India 3. Department of Mathematics,KLS, Vishwanathrao Deshpande Rural, Institute of Technology, Haliyal - 581 329, India E-mail: revankards@gmail.com, priyanka18hande@gmail.com, handesp1313@gmail.com, vijayteli22@gmail.com
Abstract: Graonac defined the second ABC index as ABC2 (G) =
X
vi vj ∈E(G)
s
1 1 2 + − . ni nj ni nj
Dae Won Lee defined the third ABC index as ABC3 (G) =
X
vi vj ∈E(G)
s
1 1 2 + − ei ej ei ej
and studied lower and upper bounds. In this paper, we defined a new index which is called third ABC Coindex and it is defined as s X 1 2 1 ABC3 (G) = + − ni nj ni nj vi vj ∈E(G) /
and we found some lower and upper bounds on ABC3 (G) index.
Key Words: Molecular graph, the third atom - bond connectivity (ABC3 ) index, the third atom - bond connectivity co-index (ABC3 ).
AMS(2010): 05C40, 05C99. §1. Introduction The topological indices plays vital role in chemistry, pharmacology etc [1]. Let G = (V, E) be a simple connected graph with vertex set V (G) = {v1 , v2 , · · · , vn } and the edge set E(G), with |V (G)| = n and |E(G)| = m. Let u, v ∈ V (G) then the distance between u and v is denoted by d(u, v) and is defined as the length of the shortest path in G connecting u and v. The eccentricity of a vertex vi ∈ V (G) is the largest distance between vi and any other vertex vj of G. The diameter d(G) of G is the maximum eccentricity of G and radius r(G) of G is the minimum eccentricity of G. 1 Received
May 17, 2017, Accepted November 19, 2017.
85
Some Lower and Upper Bounds on the Third ABC Co-index
The Zagreb indices have been introduced by Gutman and Trinajstic [2]-[5]. They are defined as, X X M1 (G) = di 2 , M2 (G) = di dj . vi ∈V (G)
vi vj ∈V (G)
The Zagreb co-indices have been introduced by Doslic [6], X
M1 (G) =
(d2i + dj 2 ),
M2 (G) =
vi vj ∈E(G) /
X
(di dj ).
vi vj ∈E(G) /
Similarly Zagreb eccentricity indices are defined as E1 (G) =
X
ei 2 ,
E2 (G) =
vi ∈V (G)
X
ei ej .
vi vj ∈V (G)
Estrada et al. defined atom bond connectivity index [7-10] s
1 2 1 + − di dj di dj
s
1 1 2 + − , ni nj ni nj
X
ABC(G) =
vi vj ∈E(G)
and Graovac defined second ABC index as ABC2 (G) =
X
vi vj ∈E(G)
which was given by replacing di , dj to ni , nj where ni is the number of vertices of G whose distance to the vertex vi is smaller than the distance to the vertex vj [11-14]. Dae and Wan Lee defined the third ABC index [16] ABC3 (G) =
X
vi vj ∈E(G)
s
1 1 2 + − . ei ej ei ej
In this paper, we have defined the third ABC co - index; ABC3 (G) as ABC3 (G) =
X
vi vj ∈E(G) /
s
1 1 2 + − ei ej ei ej
found some lower and upper bounds on ABC3 (G).
§2. Lower and Upper Bounds on ABC3 (G) Index Calculation shows clearly that (i) ABC3 (Kn ) = 0;
1 n (ii) ABC3 (K1,n−1 ) = ; 2 2
86
Deepak S. Revankar, Priyanka S. Hande, Satish P. Hande and Vijay Teli
√ (iii) ABC3 (C2n ) = 2(n − 3) n − 2; r 4n − 12 (iv) ABC3 (C2n+1 ) = n(n − 3) . (n − 1)2 1 , E2 (G)
Theorem 2.1 Let G be a simple connected graph. Then ABC3 (G) ≥ √
where E2 (G) is
the second zagreb eccentricity coindex.
Proof Since G ≇ Kn , it is easy to see that for every e = vi vj in E(G), ei + ej ≥ 3. By the definition of ABC3 coindex ABC3 (G)
=
X
vi vj ∈E(G) /
≥
X
vi vj ∈E(G) /
s
1 1 2 + − ei ej ei ej
1 ≥ r √ ei ej
1 P
vi vj ∈E(G) /
1 =q . ei ej E2 (G)
2
Theorem 2.2 Let G be a connected graph with m edges, radius r = r(G) ≥ 2, diameter d = d(G). Then, √ √ 2m √ 2m √ d − 1 ≤ ABC3 (G) ≤ r−1 d r with equality holds if and only if G is self-centered graph. Proof For 2 ≤ r ≤ ei , ej ≤ d, 1 2 1 + − ei ej ei ej
≥ ≥ ≥ ≥
1 1 2 2 1 1 2 + (1 − ) (as ej ≤ d, 1 − ≥ 0 = + (1 − )) ei ej ej ei d ej d 1 1 2 2 + (1 − ) (as ei ≤ d and (1 − ) ≥ 0) d d d d 1 1 2 + − d d d2 2 2 2 − 2 ≥ 2 (d − 1) d d d
with equality holds if and only if ei = ej = d. Similarly we can easily show that, 1 1 2 2 + − ≤ 2 (r − 1) ei ej ei ej r for 2 ≤ r ≤ ei , ej ≤ d with equality holding if and only if ei = ej = r.
2
The following lemma can be verified easily. Lemma 2.1 Let (a1 , a2 , · · · , an ) be a positive n-tuple such that there exist positive numbers
Some Lower and Upper Bounds on the Third ABC Co-index
87
A, a satisfying 0 < a ≤ ai ≤ A. Then, n (
n P
ai 2
i=1 n P
ai )2
≤
i=1
p 1 p ( A a + a A)2 4
A a n is an integer and q of numbers ai (A a) + 1 coincide with a and the remaining n − q of the a′i s coincide with A(6= a).
with equality holds if and only if a = A or q =
Theorem 2.3 Let G be a simple connected graph with m edges, r = r(G) ≥ 2, d = d(G). Then, v p u u 4m (r − 1)(d − 1) ABC3 (G) = t . √ √ rd( 1r r − 1 + 1d d − 1)2 E2(G) Proof By Theorem 2.2 we know that s √ √ 2√ 1 1 2 2√ d−1≤ + − ≤ r − 1, d ei ej ei ej r
vi vj ∈ / E(G).
(2.1)
Also by Lemma 2.3 we have a ≤ ai ≤ A. Let
√
2√ a= d − 1 and ai = d
s
(2.2)
1 1 2 + − , ei ej ei ej
vi vj ∈ / E(G)
and
√ 2√ A= r − 1. r in equations (2.1) and (2.2). We therefore know that n P n ai 2 1 i=1 P 2 ≤ 4 ( ai )
i.e.,
ai
A + a
r !2 a , A
P 2 ( ai ) 1 P ≥ 4 q , p a 2 n a2i A + a A
which implies that X
r
2
4n ≥ q A a
P
+
a2i 4n ≥ h√ p a 2 A A
√ a
P
+
a2i
√ √a A
P 4n a2i aA a2i i2 ≥ [A + a]2 A+a √
4n i2 ≥ h
P an
88
Deepak S. Revankar, Priyanka S. Hande, Satish P. Hande and Vijay Teli
and
s
X
vi vj ∈E(G)
2 1 1 2 + − ei ej ei ej
√ √ √ √ X 1 4n d2 d − 1 r2 r − 1 1 2 + − h√ √ i2 √ √ ei ej ei ej 2 2 v / r − 1 + r − 1 i vj ∈E(G) r d √ √ 8n X 1 1 2 rd d − 1 r − 1 + − . h√ i2 √ ei ej ei ej r−1 d−1 2 + v v ∈E(G) / i j r r d
≥ ≥
Therefore p 8n (r − 1)(d − 1) X 1 1 2 + − = i2 h√ √ ei ej ei ej r−1 d−1 + d rd r We know that
X
vi vj ∈E(G) /
p (r − 1)(d − 1) X 1 1 2 + − . h√ i2 √ ei ej ei ej r−1 d−1 2 v v + d i j r r
8n rd
1 1 2 + − ei ej ei ej
≥
1 E2 (G)
from Theorem 2.1. Thus, X
vi vj ∈E(G) /
X
vi vj ∈E(G) /
s
s
p 4m × (r − 1)(d − 1) 1 1 2 E2 (G), √ + − ≥ √ ei ej ei ej rd 1r r − 1 + d1 d − 1 2
1 1 2 + − ei ej ei ej
v u u ≥t
p 4m (r − 1)(d − 1) √ . rd r1 r − 1 + d√1d−1 E2 (G)
Theorem 2.4 Let G be a simple connected graph with n vertices and m edges. Then, 1
1 q ≤ ABC3 (G) ≤ √ 2 n2 m − nM1 (G) + M2 (G)
q 2nm2 − nM1 (G) − 2m2 .
Proof From Theorem 2.1, X
vi vj ∈E(G) /
s
1 1 2 + − ≥ r ei ej ei ej
1 P
ei ej
.
vi vj ∈E(G) /
Since ei ≤ (n − di ), we know that r
1 P
vi vj ∈E(G) /
ei ej
≥ pP
1 (n − di )(n − dj )
=
=
This completes the lower bound.
pP
1 (n2 − ndi − ndj + di dj )
1 q . mn2 − nM1 (G) + M2 (G)
2
Some Lower and Upper Bounds on the Third ABC Co-index
89
Now, since G ≇ Kn , ei ej ≥ 2 for vi vj ∈ / E(G), we get that X
vi vj ∈E(G) /
s
X p 1 1 2 1 + − ≤√ ei + ej − 2. ei ej ei ej 2 v v ∈E(G) / i j
By Cachy-Schwarz inequality, we also know that s X X p X 1 1 √ ei + ej − 2 ≤ √ 1 (ei + ej − 2). 2 v v ∈E(G) 2 v v ∈E(G) / / v v ∈E(G) / i j
i j
i j
Since ei ≤ n − di for vi ∈ V (G), we get that 1 √ 2
s
1 ≤ √ 2
X
1
vi vj ∈E(G) /
s m
X
vi vj ∈E(G) /
X
(ei + ej − 2)
(n − di + n − dj − 2)
vi vj ∈E(G) /
v u u X X 1 u X ≤ √ tm 2n − (di + dj ) − 2 1 2 vi vj ∈E(G) / vi vj ∈E(G) / vi vj ∈E(G) / r h i 1 = √ m 2nm − M1 (G) − 2m 2 q 1 = √ 2m2 n − mM1 (G) − 2m2 . 2
2
References [1] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley - VCH. Weinheim, 2000. [2] K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math., (2004), 57- 66. [3] K. C. Das and I Gutman, Some properties of second Zagreb index, MATCH Commun. Math. Comput. Chem., 52,(2004), 103 - 112. [4] K. C. Das, I Gutman and B. Zhou, New upper bounds on Zagreb indices, J.Math. Chem., 56,(2009), 514 - 521. [5] I. Gutman and K. C. Das, The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem., 50,(2004), 83 - 92. [6] T. Dovslic, Vertex weightedWeiner polynomial for composite graphs, Ars. Math. Contemp., 1,(2008), 66 - 80. [7] K.C. Das, D.W. Lee and A. Graovac, Some properties of the Zagreb eccentricity indices, Ars Math. Contemp., 6 (2013), 117 - 125. [8] D.W. Lee, Lower and upper bounds of Zagreb eccentricity indices on unicyclic graphs, Adv. Appl. Math. Sci., 12 (7) (2013), 403 - 410.
90
Deepak S. Revankar, Priyanka S. Hande, Satish P. Hande and Vijay Teli
[9] M. Ghorbani, M.A. Hosseinzadeh, A new version of Zagreb indices, Filomat, 26 (2012), 93 - 100. [10] D. Vukicevic, A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chim. Slov., 57 (2010), 524 - 528. [11] E. Estrada, L. Torres, L. Rodrguez, I. Gutman, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem., A 37 (1998), 849 - 855. [12] E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett., 463 (2008), 422 - 425. [13] K.C. Das, Atom-bond connectivity index of graphs, Discrete Appl. Math., 158 (2010), 1181-1188. [14] K.C. Das, I. Gutman, B. Furtula, On atom-bond connectivity index, Chem. Phys. Lett., 511 (2011), 452 - 454. [15] Dae Won Lee, Some lower and upper bounds on the third ABC index, AKCE International Journal of Graphs and Combinatorics, 13, (2016), 11- 15. [16] D. S. Revankar, S. P. Hande, S. R. Jog, P. S. Hande and M. M. Patil, Zagreb coindices on some graph operators of Tadpole graph, International Journal of Computer and Mathemaical Sciences, 3 (3), (2014).